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ON HARMONIC FUNCTIONS OF SYMMETRIC LEVYPROCESSES

ANTE MIMICA

Abstract. We consider some classes of Levy processes for which the estimate ofKrylov and Safonov (as in [BL02]) fails and thus it is not possible to use the stan-dard iteration technique to obtain a-priori Holder continuity estimates of harmonicfunctions. Despite the faliure of this method, we obtain some a-priori regularityestimates of harmonic functions for these processes. Moreover, we extend resultsfrom [SSV06] and obtain asymptotic behavior of the Green function and the Levydensity for a large class of subordinate Brownian motions, where the Laplaceexponent of the corresponding subordinator is a slowly varying function.

1. Introduction

Recently there has been much interest in investigation of the continuity propertiesof harmonic functions with the respect to various non-local operators. An exampleof such operator L is of the form

(Lf)(x) =

∫

Rd\0

(f(x+ h)− f(x)− 〈∇f(x), h〉1|h|≤1

)n(x, h) dh (1.1)

for f ∈ C(Rd) bounded. Here n : Rd × (Rd \ 0) → [0,∞) is a measurable functionsatisfying

c1|h|−d−α ≤ n(x, h) ≤ c2|h|

−d−α ,

for some constants c1, c2 > 0 and α ∈ [0, 2].

It is known that for α ∈ (0, 2) Holder regularity estimates hold for L-harmonicfunctions (see [BL02] for a probabilistic and [Sil06] for an analytic approach) .

2000 Mathematics Subject Classification. Primary 60J45, 60J75, Secondary 60J25.Key words and phrases. geometric stable process, Green function, harmonic function, Levy

process, modulus of continuity, subordinator, subordinate Brownian motion.Research supported in part by German Science Foundation DFG via IGK ”Stochastics and real

world models” and SFB 701.1

2 ANTE MIMICA

For α = 0, techniques developed so far are not applicable. One of our aims is toinvestigate this case using a probabilistic approach.

In many cases the operator of the form (1.1) can be understood as the infinitesimalgenerator of a Markov jump process. The kernel n(x, h) can be thought of as themeasure of intensity of jumps of the process.

Let us describe the stochastic process we are considering. Let S = (St : t ≥ 0) bea subordinator such that its Laplace exponent φ defined by φ(λ) = − log

(Ee−λS1

)

satisfies

limλ→+∞

φ′(λx)

φ′(λ)= x

α2−1 for any x > 0 (1.2)

for some α ∈ [0, 2] . Let B = (Bt,Px) be an independent Brownian motion in Rd and

define a new process X = (Xt,Px) in Rd by Xt = B(St) . It is called the subordinate

Brownian motion.

Example 1: Let S be a subordinator with the Laplace exponent φ satisfying

limλ→+∞

φ(λ)

λα/2ℓ(λ)= 1

with α ∈ (0, 2) and ℓ : (0,∞) → (0,∞) that varies slowly at infinity (i.e. for any

x > 0, ℓ(λx)ℓ(λ)

→ 1 as λ→ +∞).

Then (1.2) holds (we just use Theorem 1.7.2 in [BGT87] together with the fact that

φ′ is decreasing and φ(λ) =∫ λ

0φ′(t) dt) and

n(x, h) = n(h) ≍ |h|−d−αℓ(|h|−2), |h| → 0+ ,

which means that n(h)|h|−d−αℓ(|h|−2)

stays between two positive constants as |h| → 0+ .

Choosing ℓ ≡ 1 we see that the rotationally invariant α-stable process (whose in-finitesimal generator is the fractional Laplacian L = −(−∆)

α2 ) is included in this

class. Other choices of ℓ allow us to consider processes which are not invariant undertime-space scaling.

The processes described in Example 1 with ℓ ≡ 1 belong to the class of Levy stableor, more generally, stable-like Markov jump processes. The potential theory ofsuch processes is well investigated (see [BL02, CK03, SV04, BS05, BK05, RSV06,KS07, CK08, Mim10, Szt10, KM11]). For example, it is known that harmonicfunctions of such processes satisfy Holder regularity estimates and the scale invariantHarnack inequality holds. Also, two sided heat kernel estimates are obtained forthese processes.

HARMONIC FUNCTIONS OF LEVY PROCESSES 3

Not much is known about harmonic functions in the case when the correspondingsubordinators belong to ’boundary’ cases, i.e. α ∈ 0, 2 (see [SSV06, Mim11,Mim12]). For the class of geometric stable processes only the non-scale invariantHarnack inequality was proved and on-diagonal heat kernel upper estimate is notfinite (see [SSV06]).

The following two examples belong to these ’boundary’ cases and are covered by ourapproach.

Example 2: Let φ(λ) = log(1 + λ). The corresponding process X is known as thevariance gamma process. In (1.2) we have α = 0 . Moreover, it will be proved (seeTheorem 4.1) that

n(x, h) = n(h) ≍ |h|−d, |h| → 0 + . (1.3)

This example can be generalized in various ways. For example, we can take k ∈ N

and consider φk = φ . . . φ︸ ︷︷ ︸

k times

. Then (see Theorem 4.1 or Subsection 7.1)

nk(x, h) = nk(h) ≍ |h|−d

log · · · log︸ ︷︷ ︸

k−1 times

1|h| · . . . · log log

1|h| · log

1|h|

−1

, |h| → 0 + .

Another generalization is to consider φ(λ) = log(1 + λβ/2) for some β ∈ (0, 2]. Theprocess X is known as the β-geometric stable process and the behavior of n is givenalso by (1.3).

Example 3: Let φ(λ) = λlog(1+

√λ)

. Then in (1.2) we have α = 2 and (see Theorem

4.1)

n(x, h) = n(h) ≍ |h|−d−2(

log 1|h|

)−2

, |h| → 0 + .

This behavior shows that small jumps of this process have higher intensity thansmall jumps of any stable process.

A measurable bounded function f : Rd → R is said to be harmonic in an open setD ⊂ R

d if for any relatively compact open set B ⊂ B ⊂ D

f(x) = Exf(XτB) for any x ∈ B,

where τB = inft > 0: Xt 6∈ B .

The main theorem is the following regularity result, which covers cases α ∈ [0, 1).The novelty of this result rests on the case α = 0. By Br(x0) we denote the openball with center x0 ∈ R

d and radius r > 0 .

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Theorem 1.1. Let S be a subordinator such that its Levy and potential measureshave decreasing densities. Assume that the Laplace exponent of S satisfies (1.2) withα ∈ [0, 1). Let X be the corresponding subordinate Brownian motion and let d ≥ 3.

There is a constant c > 0 such that for any r ∈ (0, 14) and any bounded function

f : Rd → R which is harmonic in B4r(0),

|f(x)− f(y)| ≤ c‖f‖∞φ (r−2)

φ (|x− y|−2)for all x, y ∈ B r

4(0) .

Applying Theorem 1.1 to Example 1, we obtain expected Holder regularity esti-mates. Within this example the result is new when the scaling is lost, e. g.

φ(λ) = λα2 [log(1 + λ)]1−

α2 .

The situation is more interesting in Example 2, e.g. for the geometric stable process.For this process we obtain logarithmic regularity estimates:

|f(x)− f(y)| ≤ c‖f‖∞ log(r−1)1

log(|x− y|−1),

It is still unknown whether Holder regularity estimates hold for harmonic functionsof this process (or, generally, of the processes belonging to the case α = 0).

Let us explain why known analytic and probabilistic techniques do not work in thecase α = 0. The main idea in the proof of the a priori Holder estimates of harmonicfunctions relies on the estimate of Krylov and Safonov.

In probabilistic setting this estimate can be formulated as follows. There is a con-stant c > 0 such that for every closed subset A ⊂ Br(0) and x ∈ B r

2(0)

Px(TA < τB(0,r)) ≥ c|A|

|Br(0)|, (1.4)

where TA = τAc is the first hitting time of A and |A| denotes the Lebesgue measureof the set A .

Performing a computation similar to the one in the proof of Proposition 3.4 in [BL02](see also Lemma 3.4 in [SV04]) we deduce

Px(TA < τBr(0)) ≥ cr−2φ′(r−2)

φ(r−2)

|A|

|Br(0)|.

HARMONIC FUNCTIONS OF LEVY PROCESSES 5

If α ∈ (0, 2), it can be seen that r−2φ′(r−2)φ(r−2)

≍ 1 as r → 0+ . This gives estimate

of the form (1.4) and thus the standard Moser’s iteration procedure for obtaininga-priori Holder regularity estimates of harmonic functions can be applied (see theproof of Theorem 4.1 in [BL02] for a probabilistic version).

The situation is quite different for α = 0. To find a counterexample we will use thefollowing result.

Proposition 1.2. Let S be as subordinator such that its Levy and potential measureshave decreasing densities and whose Laplace exponent satisfies (1.2) with α ∈ [0, 1).Let X be the corresponding subordinate Brownian motion and let d ≥ 3.

There is a constant c > 0 such that for every r ∈ (0, 1) and x ∈ B r4(0)

Px(XτBr2(0)

∈ Br(0) \B r2(0)) ≤ c

r−2φ′(r−2)

φ(r−2).

For α = 0 it will follow that limr→0+

r−2φ′(r−2)φ(r−2)

= 0 (see (2.10)). Therefore in this case

(1.4) does not hold, since

limr→0+

P0(TBr(0)\B r4(0) < τBr(0)) ≤ lim

r→0+P0(XτBr

4(0)

∈ Br(0) \B r4(0)) = 0 .

Considering process X in the setting of metric measure spaces (as in [CK08] or[BGK09]) the news feature appears. Theorem 4.1 shows that the jumping kernel ofthe process X is of the form n(x, h) = j(|h|) with

j(r) ≍r−2φ′(r−2)

φ(r−2)·E0τBr(0)

|Br(0)|, r → 0 + .

In the case α = 0 the term r−2φ′(r−2)φ(r−2)

becomes significant. This has not yet been

treated within this framework.

The latter discussion shows that the question of the continuity of harmonic functionsbecomes interesting even in the case when the kernel n(x, h) is space homogeneous,or in other words, in the case of a Levy process. There is no known technique thatcovers this situation in the case of a more general jump process.

Our technique relies on asymptotic properties of the underlying subordinator. Thepotential density can be analyzed using the de Haan theory of slow variation (see[BGT87]).

6 ANTE MIMICA

On the other hand, there is no known Tauberian theorem that can be applied toobtain asymptotic behavior of the Levy density µ of the subordinator. For thispurpose we perform asymptotic inversion of the Laplace transform (see Proposition3.2) to get

µ(t) ≍ t−2φ′(t−2), t→ 0 + .

These techniques allow us to extend results from [SSV06] to much wider class ofsubordinators whose Laplace exponents are logarithmic or, more generally, slowlyvarying functions.

Although we do not obtain regularity estimates of harmonic functions for cases whenα ∈ [1, 2], it is possible to say something about the behavior of the jumping kerneland the Green function. In this sense, the case α = 2 is also new. For examplethe Green function of the process corresponding to the Example 3 above has thefollowing behavior:

G(x, y) ≍ |x− y|2−d log(|x− y|−1), |x− y| → 0 + .

We may say that such process X is ’between’ any stable process and Brownianmotion.

Let us briefly comment the technique we are using to prove the regularity result. InSection 2 it will be seen that any bounded function f which is harmonic in B2r(0)can be represented as

f(x) =

∫

Br(0)cKBr(0)(x, z)f(z) dz, x ∈ Br(0) ,

where KBr(0)(x, z) is the Poisson kernel of the ball Br(0).

The following estimate of differences of Poisson kernel is the key to the proof ofTheorem 1.1:

|KBr(0)(x1, z)−KBr(0)(x2, z)| ≤

c |z|−d φ((|z|−r)−2)φ(|x−y|−2)

r < |z| ≤ 2rj( |z|

2)

φ(|x−y|−2)|z| > 2r

for x1, x2 ∈ B r8(0) (see Proposition 5.3).

Similar type estimate has been obtained in [Szt10] for stable Levy processes usingscaling argument and the explicit behavior of the transition density. In our settingthere are many cases where the behavior of the transition density is not known andthe scaling argument does not work. Our idea is to establish the following Green

HARMONIC FUNCTIONS OF LEVY PROCESSES 7

function difference estimates:

|G(x1, y)−G(x2, y)| ≤ cr−2φ′(r−2)

φ(r−2)2rd

(

1 ∧ |x−y|r

)

for all y ∈ Rd and x1, x2 6∈ Br(y) (see Proposition 5.1).

The paper is organized as follows. In Section 2 we introduce all concepts we needthroughout the paper. Section 3 is devoted to the study of subordinators. We obtainasymptotic properties of Levy and potential densities. In Section 4 asymptoticalproperties of the Green function and Levy density of the subordinate Brownianmotions are obtained. Difference estimates of the Green function and the Poissonkernel are the main subject of Section 5. This type of estimates are the mainingredient in the proof of the regularity result in Section 6. In Section 7 we applyour results to some new examples.

Notation. For two functions f and g we write f ∼ g if f/g converges to 1 andf ≍ g if f/g stays between two positive constants. The n-th derivative of f (ifexists) is denoted by f (n).

The logarithm with base e is denoted by log and we introduce the following notationfor iterated logarithms: log1 = log and logk+1 = log logk for k ∈ N.

We say that f : R → R is increasing if s ≤ t implies f(s) ≤ f(t) and analogously fora decreasing function.

The standard Euclidian norm and the standard inner product in Rd are denoted

by | · | and 〈·, ·〉, respectively. By Br(x) = y ∈ Rd : |y − x| < r we denote the

open ball centered at x with radius r > 0. The Gamma function is defined byΓ(ρ) =

∫∞0tρ−1e−t dt for ρ > 0.

2. Preliminaries

2.1. Levy processes and their potential theory. A stochastic process X =(Xt : t ≥ 0) with values in R

d (d ≥ 1) defined on a probability space (Ω,F ,P) is saidto be a Levy process if it has independent and stationary increments, its trajectoriesare P-a.s. right continuous with left limits and P(X0 = 0) = 1 .

The characteristic function of Xt is always of the form

E exp i〈ξ,Xt〉 = exp −tΦ(ξ),

8 ANTE MIMICA

where Φ is called the characteristic (or Levy) exponent of X . It has the followingLevy-Khintchine representation

Φ(ξ) = i〈γ, ξ〉+1

2〈Aξ, ξ〉+

∫

Rd

(1− ei〈x,ξ〉 + i〈x, ξ〉1|x|≤1

)Π(dx).

Here γ ∈ Rd, A is a non-negative definite symmetric d × d real matrix and Π is a

measure on Rd, called the Levy measure of X , satisfying

Π(0) = 0 and

∫

Rd

(1 ∧ |x|2)Π(dx) <∞ .

The Brownian motion B = (Bt : t ≥ 0) in Rd with transition density p0(t, x, y) =

(4πt)−d/2 exp

− |x−y|24t

is an example of a Levy process with the characteristic ex-

ponent Φ(ξ) = |ξ|2 .

A subordinator is a stochastic process S = (St : t ≥ 0) which is a Levy process in R

such that St ∈ [0,∞) for every t ≥ 0. In this case it is more convenient to considerthe Laplace transform of St:

E exp −λSt = exp −tφ(λ), λ > 0.

The function φ : (0,∞) → (0,∞) is called the Laplace exponent of S and it has thefollowing representation

φ(λ) = γλ+

∫

(0,∞)

(1− e−λt)µ(dt).

Here γ ≥ 0 and µ is also called the Levy measure of S and it satisfies the followingintegrability condition:

∫

(0,∞)(1 ∧ t)µ(dt) <∞.

The potential measure of the subordinator S is defined by

U(A) = E

[∫ ∞

0

1St∈A dt

]

for a measurable A ⊂ [0,∞) .

The Laplace transform of U is then

LU(λ) :=

∫

(0,∞)

e−λt U(dt) =1

φ(λ). (2.1)

Assume that the processes B and S just described are independent. We define anew stochastic process X = (Xt : t ≥ 0) by Xt = B(St) and call it the subordinate

HARMONIC FUNCTIONS OF LEVY PROCESSES 9

Brownian motion. It is a Levy process with the characteristic exponent Φ(ξ) =φ(|ξ|2) and the Levy measure of the form Π(dx) = j(|x|) dx with

j(r) =

∫

(0,∞)

(4πt)−d/2 exp

−r2

4t

µ(dt) . (2.2)

The process X has the transition density and it is given by

p(t, x, y) =

∫

[0,∞)

(4πs)−d/2 exp

−|x− y|2

4s

P(St ∈ ds) . (2.3)

When X is transient, we can define the Green function of X by

G(x, y) =

∫

(0,∞)

p(t, x, y) dt, x, y ∈ Rd, x 6= y .

The Green function can be considered as the density of the Green measure definedby

G(x,A) = Ex

[∫ ∞

0

1Xt∈A dt

]

, A ⊂ Rd measurable ,

since G(x,A) =∫

AG(x, y) dy.

Using (2.3) we can rewrite it as G(x, y) = g(|y − x|) with

g(r) =

∫

(0,∞)

(4πt)−d/2 exp

−r2

4t

U(dt) . (2.4)

Let D ⊂ Rd be a bounded open set. We define the process killed upon exiting D

XD = (XDt : t ≥ 0) by

XDt =

Xt t < τD∂ t ≥ τD,

where ∂ is an extra point adjoined to D.

Using the strong Markov property we can see that the Green measure of XD is

GD(x,A) = Ex

[∫ ∞

0

1XDt ∈A dt

]

= Ex

[∫ ∞

0

1Xt∈A dt

]

− Ex

[∫ ∞

τD

1Xt∈A dt

]

= G(x,A)− Ex[G(XτD , A); τD <∞] .

10 ANTE MIMICA

Thus in the transient case the Green function of XD can be written as

GD(x, y) = G(x, y)− Ex[G(XτD , y); τD <∞], x, y ∈ D, x 6= y .

Since X is, in particular, an isotropic Levy process it follows from [Szt00] that

Px(XτBr(0)∈ ∂Br(0)) = 0, x ∈ Br(0) .

for any r > 0 and x ∈ Br(0) . This allows us to use the Ikeda-Watanabe formula(see Theorem 1 in [IW62]):

Px(XτBr(0)∈ F ) =

∫

F

∫

Br(0)

GBr(0)(x, y)j(|z − y|) dy dz , (2.5)

for x ∈ Br(0) and F ⊂ Br(0)c.

Defining a function KBr(0) : Br(0)× Br(0)c→ [0,∞) by

KBr(0)(x, z) =

∫

Br(0)

GBr(0)(x, y)j(|z − y|) dy (2.6)

the Ikeda-Watanabe formula (2.5) reads

Px(XτBr(0)∈ F ) =

∫

F

KBr(0)(x, z) dz . (2.7)

The function KBr(0) will be called the Poisson kernel for the ball Br(0) .

2.2. Bernstein functions and subordinators. A function φ : (0,∞) → (0,∞) issaid to be a Bernstein function if φ ∈ C∞(0,∞) and (−1)nφ(n) ≤ 0 for all n ∈ N.Every Bernstein φ function has the following representation:

φ(λ) = γ1 + γ2λ+

∫

(0,∞)

(1− e−λt)µ(dt), (2.8)

where γ1, γ2 ≥ 0 and µ is a measure on (0,∞) satisfying∫

(0,∞)(1 ∧ t)µ(dt) <∞.

Using the elementary inequality ye−y ≤ 1 − e−y, y > 0 we deduce from (2.8) thatevery Bernstein function φ satisfies:

λφ′(λ) ≤ φ(λ) for any λ > 0 . (2.9)

There is a strong connection between subordinators and Bernstein functions. To bemore precise, φ is a Bernstein function such that φ(0+) = 0 (i.e. γ1 = 0 in (2.8)) ifand only if it is the Laplace exponent of some subordinator. If φ(0+) > 0, φ can beunderstood as the Laplace exponent of a subordinator killed with rate φ(0+) (seeChapter 3 in [Ber96]).

HARMONIC FUNCTIONS OF LEVY PROCESSES 11

A Bernstein function φ is a complete Bernstein function if the Levy measure in (2.8)has a completely monotone density, i.e. µ(dt) = µ(t) dt, where µ : (0,∞) → (0,∞),µ ∈ C∞(0,∞) and (−1)nµ(n) ≥ 0 for any n ∈ N .

Let us mention some properties of complete Bernstein function (see [SSV10]). Thecomposition of two complete Bernstein function is a complete Bernstein functionand if φ is a complete Bernstein function, then φ⋆(λ) = λ

φ(λ)is also a complete

Bernstein function.

Assume that S is a subordinator with the Laplace exponent φ and infinite Levymeasure. Then φ⋆ is a Bernstein function if and only if the potential measure Uhas a decreasing density u with respect to the Lebesgue measure. Moreover, if νdenotes the Levy measure of the subordinator with the Laplace exponent φ⋆, thenu(t) = ν(t,∞) for any t > 0 .

2.3. Regular variation. A function f : (0,∞) → (0,∞) varies regularly (at infin-ity) with index ρ ∈ R if

limλ→+∞

f(λx)

f(λ)= xρ for every x > 0 .

If ρ = 0, then we say that f is slowly varying. Regular (slow) variation at 0 isdefined similarly.

If f varies regularly with index ρ ∈ R, then there exists a slowly varying function ℓso that f(λ) = λρℓ(λ) .

Let ℓ be a slowly varying function such that L(λ) =∫ λ

0ℓ(t)tdt exists for all λ > 0 .

Then L is slowly varying,

limλ→+∞

L(λ)

ℓ(λ)= +∞ (2.10)

and

limλ→+∞

L(λx)− L(λ)

ℓ(λ)= log x for every x > 0 .

(see Proposition 1.5.9 a and p. 127 in [BGT87]).

12 ANTE MIMICA

3. Subordinators

Let S = (St : t ≥ 0) be a subordinator with the Laplace exponent φ satisfying thefollowing conditions:

(A-1) there is α ∈ [0, 2] such that φ′ varies regularly at infinity with index α2− 1,

i.e.

limλ→+∞

φ′(λx)

φ′(λ)= x

α2−1 for every x > 0 ,

(A-2) the Levy measure is infite and has a decreasing density µ ,(A-3) the potential measure has a decreasing density u .

When α = 2 we additionaly assume:

(A-4) λ 7→(

λφ(λ)

)′varies regularly at infinity with index −1 .

Remark 3.1. (a) The most important assumption is (A-1) (and (A-4) when α =2). Other assumptions hold for a large class of subordinators (e. g. when φis a complete Bernstein function with infinite Levy measure).

(b) By Karamata’s theorem (see Theorem 1.5.11 in [BGT87]) it follows from(A-1) that φ varies regularly at infinity with index α

2.

Proposition 3.2. Let α ∈ [0, 2) and let S be a subordinator satisfying (A-1) and(A-2). Then

µ(t) ≍ t−2φ′(t−1), t→ 0 + .

Proof. Let ε > 0. By a change of variable

φ(λ+ ε)− φ(λ) =

∫ ∞

0

(e−λt − e−(λ+ε)t)µ(t) dt

= λ−1

∫ ∞

0

e−t(1− e−ελ−1t)µ(λ−1t) dt . (3.1)

Since µ is decreasing, the following holds

φ(λ+ ε)− φ(λ) ≥ λ−1µ(λ−1)

∫ 1

0

e−t(1− e−ελ−1t) dt .

Now we can apply Fatou lemma to deduce

φ′(λ) = limε→0+

φ(λ+ ε)− φ(λ)

ε≥ λ−2µ(λ−1)

∫ 1

0

te−t dt

= λ−2µ(λ−1)(1− 2e−1) .

HARMONIC FUNCTIONS OF LEVY PROCESSES 13

By setting λ = t−1 we get the upper bound

µ(t) ≤t−2f ′(t−1)

1− 2e−1for every t > 0 . (3.2)

Now we prove the lower bound. Using (3.1), for any r ∈ (0, 1) and ε > 0 we canwrite

φ(λ+ ε)− φ(λ) = I1 + I2 (3.3)

with

I1 = λ−1

∫ r

0

e−t(1− e−ελ−1t)µ(λ−1t) dt

I2 = λ−1

∫ ∞

r

e−t(1− e−ελ−1t)µ(λ−1t) dt .

Since µ is decreasing, the dominated convergence theorem yields

lim supε→0+

I2ε

≤ λ−2µ(λ−1r)

∫ ∞

r

te−t dt

= (r + 1)e−rλ−2µ(λ−1r) . (3.4)

To handle I1 first we use the theorem of Potter (see Theorem 1.5.6 (iii) in [BGT87])to conclude that there are constants c1 > 0 and δ > 0 such that

φ′(λt−1)

φ′(λ)≤ c1t

δ for all λ ≥ 1 and t ≤ 1 . (3.5)

Therefore, by (3.2), (3.5) and the dominated convergence theorem

lim supε→0+

I1ε

≤ lim supε→0+

1

1− 2e−1

∫ r

0

e−t 1− e−ελ−1t

ελ−1tt−1φ′(λt−1) dt

≤ φ′(λ)c1

1− 2e−1

∫ r

0

tδ−1e−t dt . (3.6)

Combining (3.3), (3.4) and (3.6) we deduce

φ′(λ) ≤ φ′(λ)c1

1− 2e−1

∫ r

0

tδ−1e−t dt+ (r + 1)e−rλ−2µ(λ−1r) .

Furthermore, by choosing r ∈ (0, 1) so that c11−2e−1

∫ r

0tδ−1e−t dt ≤ 1

2we get

µ(λ−1r) ≥er

2(r + 1)λ2φ′(λ) for every λ ≥ 1 .

14 ANTE MIMICA

By (A-1) we can find t0 ∈ (0, r) so that φ′(rt−1)φ′(t−1)

≥ rα2 −1

2for any t ∈ (0, t0). The lower

bound nwo follows:

µ(t) ≥r1+

α2 er

4(r + 1)t−2φ′(t−1) for every t ∈ (0, t0) .

Remark 3.3. The precise asymptotical behavior of µ when α ∈ (0, 2) can be obtainedby Karamata’s Tauberian theorem. This is not the case when α = 0.Under an additional assumption

t 7→ taatµ(t) is monotone on (0, T ) for some a ≥ 0 and T > 0,

it is possible to prove the following precise asymptotics in the case α = 0 :

µ(t) ∼ t−2φ′(t−1), t→ 0 + .

Proposition 3.4. Let α ∈ [0, 2) and let S be a subordinator satisfying (A-1) and(A-3). Then

u(t) ∼1

Γ(1− α

2

)t−2φ′(t−1)

φ(t−1)2, t→ 0 + .

Remark 3.5. It can be proved that

u(t) ≍1

Γ(1− α

2

)t−2φ′(t−1)

φ(t−1)2, t→ 0 + .

similarly as in Proposition 3.2. It is enough to note

ψ(λ+ ε)− ψ(λ) =

∫ ∞

0

(e−λt − e−(λ+ε)t)u(t) dt

with ψ(λ) = − 1φ(λ)

.

The main reason why we need precise asymptotics of u is to be able to handle thecase α = 2 by duality.

Proof of Proposition 3.4. Let us first consider the case α = 0. In this case ℓ(λ) =λφ′(λ) varies slowly (at infinity) and thus it follows from Subsection 2.3 that φ(λ) =∫ λ

0ℓ(t)tdt also varies slowly and

limλ→+∞

φ(λx)− φ(λ)

λφ′(λ)= log x for every x > 0 .

HARMONIC FUNCTIONS OF LEVY PROCESSES 15

This and (2.1) imply

LU(

1λx

)− LU

(1λ

)

1λφ′(1λ

)φ(1λ

)2 =φ(1λ

)− φ

(1λx

)

1λφ′(1λ

)φ(1λ

)

φ(

1λx

) → log x, λ→ 0+

for any x > 0. Now we can apply de Haan’s Tauberian theorem (see 0 –version of[BGT87, Theorem 3.9.1]) to deduce

U(λx)− U(λ)1λφ′(1λ

)φ(1λ

)2 → log x, λ→ 0 + .

If we apply de Haan’s monotone density theorem (see [BGT87, Theorem 3.6.8]) wefinally obtain

u(t) ∼t−2φ′ (t−1)

tφ (t−1)2, t→ 0 + .

The case α ∈ (0, 2) is already known. We give the proof for the sake of completenessand adapt the result to the formula obtained in the case α = 0 .

Since

LU(λ) =1

φ(λ)

varies regularly at infinity with index −α2, Karamata’s Tauberian theorem (see The-

orem 1.7.1 in [BGT87]) implies

U([0, t]) ∼1

Γ(1− α

2

)1

φ (t−1), t→ 0 + .

Then by Karamata’s monotone density theorem (see Theorem 1.7.2 in [BGT87]) wededuce

u(t) ∼α

2Γ(1− α

2

)1

tφ (t−1)∼

1

Γ(1− α

2

)t−2φ′(t−1)

φ (t−1)2, t→ 0 + .

Now we consider the case α = 2.

Proposition 3.6. Let α = 2 and let S be a subordinator satisfying (A-1), (A-2)and (A-4). Then

µ(t) ∼ t−2(tφ(t−1)− φ′(t−1)

), t→ 0 + .

16 ANTE MIMICA

Proof. Since potential density exists by (A-3) we see that φ it follows that φ⋆(λ) =λ

φ(λ)defines the Laplace exponent of a (possibly killed) subordinator, which we de-

note by T (see Section 2).

Note that the subordinator T corresponds to the case of α = 0. If we denotepotential density of T by v, then

v(t) = µ(t,∞), t > 0 .

Proposition 3.4 yields∫ ∞

t

µ(s) ds ∼t−2(φ⋆)′(t−1)

φ⋆(t−1)2, t→ 0 + . (3.7)

By assumption (A-4) we know that t 7→ (φ⋆)′(t−1) varies regularly at 0 with index

1 and thus t 7→ t−2(φ⋆)′(t−1)φ⋆(t−1)2

varies regularly at 0 with index −1.

Now we change variable in the integral on the left-hand side in (3.7) and conclude

∫ t−1

0

µ(s−1)ds

s2∼t−2(φ⋆)′(t−1)

(φ⋆(t−1))2, t→ 0 + .

This gives (r = t−1):∫ r

0

µ(s−1)ds

s2∼r2(φ⋆)′(r)

φ⋆(r)2, r → ∞ .

Note that the right-hand side is now regularly varying at infinity with index 1 andthus by Karamata’s monotone density theorem (see Theorem 1.7.2 in [BGT87]) wededuce

µ(r−1)

r2∼r(φ⋆)′(r)

φ⋆(r)2.r → ∞ ,

Going back (t = r−1) we conclude

µ(t) ∼t−3(φ⋆)′(t−1)

φ⋆(t−1)2, t→ 0 + .

Proposition 3.7. Let α = 2 and let S be a subordinator satisfying (A-1) and (A-3).Then the following is true

u(t) ∼1

φ′(t−1)∼

1

tφ(t−1), t→ 0 + .

HARMONIC FUNCTIONS OF LEVY PROCESSES 17

Proof. By (2.1) we get

LU(λ) =1

φ(λ)∼

1

λφ′(λ), λ→ +∞

and thus by Karamata’s Tauberian theorem (see Theorem 1.7.1 in [BGT87]) itfollows that

U([0, t]) ∼1

Γ(2)

t

φ′(t−1), t→ 0+

since λ 7→ λφ′(λ) varies regualrly at infinity with index 1. By applying Karamata’smonotone density (see Theorem 1.7.2 in [BGT87]) theorem we deduce

u(t) ∼1

φ′(t−1), t→ 0 + .

4. Levy density and Green function

Let S be a subordinator as in Section 3 and let X be the corresponding subordinateBrownian motion in R

d with d ≥ 3. Our aim is to establish asymptotical behaviorof the Levy density and Green function of X .

Recall that the Levy density of X is of the form j(|x|), where j is given by (2.2).

Theorem 4.1. Assume that S satisfies (A-1) with some α ∈ [0, 2] and (A-2). Ifα ∈ [0, 2), then

j(r) ≍ r−d−2φ′(r−2), r → 0 + .

If α = 2 and (A-4) holds, then

j(r) ≍ r−d−2(r2φ(r−2)− φ′(r−2)

), r → 0 + .

Proof. This result follows directly from Proposition 3.2 and Proposition 3.6 togetherwith Lemma A.1, where a = 1

4, b = 1 + α

2.

For α ∈ [0, 2) the slowly varying function is given by ℓ(t) = tα2−1φ′(t−1). When

α = 2 we take

ℓ(t) = tφ(t−1)− φ′(t−1) =t (φ⋆(t−1))

′

φ⋆(t−1)2,

18 ANTE MIMICA

which varies slowly, since φ⋆(λ) = λφ(λ)

is slowly varying by (A-1) and Karamata’s

Theorem (see Theorem 1.5.11 in [BGT87]) and has a derivative that varies regularlywith index 1 by (A-4).

The Green function of X is of the form G(x, y) = g(|y − x|), where g is given by(2.4).

Theorem 4.2. Assume that S satisfies (A-1) with some α ∈ [0, 2] and (A-3). Letd ≥ 3. If α ∈ [0, 2), then

g(r) ≍ r−d−2 φ′(r−2)

φ(r−2)2, r → 0 + .

If α = 2, then

g(r) ≍ r−d+2 1

φ′(r−2)≍ r−d 1

φ(r−2), r → 0 + .

Proof. We use Lemma A.1 with a = 14, b = 1 − α

2, Proposition 3.4 and Proposition

3.7.

When α ∈ [0, 2) we define ℓ(t) = t−1φ′(t−1)φ(t−1)

which varies slowly at 0 .

In the case α = 2, we let ℓ(t) = 1φ′(t−1)

or ℓ(t) = 1tφ(t−1)

which both vary slowly at

0.

Using asymptotical results from this section we can now prove the proposition thatgives a counterexample for the estimate of the Krylov and Safonov.

Proof of Proposition 1.2. By (2.5),

Px(XτBr4(0)

∈ Br(0) \B r4(0)) =

∫

Br(0)\B r4(0)

∫

B r4(0)

GB r4(x, y)j(|z − y|) dy

= I1 + I2 .

HARMONIC FUNCTIONS OF LEVY PROCESSES 19

Using Theorems 4.1 and 4.2 it follows that

I1 =

∫

Br(0)\B r4(0)

∫

B 3r16

(0)

GB r4(x, y)j(|z − y|) dy dz

≤ j( r16)|Br(0) \B r

4(0)|

∫

Br(0)

g(|y|) dy dz

≤ c1r−2φ′(r−2)

φ(r−2).

On the other hand,

I2 =

∫

Br(0)\B r4(0)

∫

B r4(0)\B 3r

16(0)

GB r2(x, y)j(|z − y|) dy dz

≤ g( r16)

∫

Br(0)\B r4(0)

∫

B r4(z)

j(|y|) dy dz . (4.1)

To estimate the inner integral, note that B r4(z) ⊂ B1(0) \ B|z|− r

4(0) for any z ∈

Br(0) \B r4(0) and so, by Theorem 4.1,

∫

B r4(z)

j(|y|) dy ≤ c2

∫ 1

|z|− r4

s−3φ′(s−2) ds ≤ c3φ((|z| −r4)−2) . (4.2)

Thus, by Theorem 4.2 and (4.1)

I2 ≤ c4r−d−2 φ

′(r−2)

φ(r−2)2

∫ r

r4

φ((t− r4)−2)td−1 dt

≤ c4r−3 φ

′(r−2)

φ(r−2)2

∫ r4

0

φ(s−2) ds

≤ c5r−2φ′(r−2)

φ(r−2).

In the last equality we have used Karamata’s theorem (see Theorem 1.5.11 in[BGT87]) and the fact that α ∈ [0, 1) .

20 ANTE MIMICA

5. Difference estimates

Let X be the stochastic process in Rd as in Section 4 and assume that d ≥ 3. In

particular X is transient.

In this section we prove the difference estimates of the Green function and thePoisson kernel.

Although we are slightly abusing notation, we set G(x) := G(0, x) = g(|x|).

Proposition 5.1. There is a constant c > 0 such that for every r ∈ (0, 1)

|G(x)−G(y)| ≤ cg(r)(

1 ∧ |x−y|r

)

for all x, y 6∈ Br(0) .

Proof. Assume first that |x− y| < r2. By the mean value theorem it follows that for

any t > 0 there exists ϑ = ϑ(x, y, t) ∈ [0, 1] such that∣∣∣∣e−

|x|2

4t − e−|y|2

4t

∣∣∣∣≤ |x+ϑ(y−x)|

2te−

|x+ϑ(y−x)|2

4t |x− y|

≤ 2 |x−y|√te−

|x+ϑ(y−x)|2

8t ,

where in the last line the following elementary inequality was used

se−s2 < 2e−s2

2 , s > 0 .

Then |x+ ϑ(y − x)| ≥ |x| − ϑ|y − x| ≥ r2implies

∣∣∣∣e−

|x|2

4t − e−|y|2

4t

∣∣∣∣≤ 2 |x−y|√

te−

r2

32t . (5.1)

By (5.1)

|G(x)−G(y)| ≤ (4π)−d/2

∫ ∞

0

t−d/2

∣∣∣∣e−

|x|2

4t − e−|y|2

4t

∣∣∣∣u(t) dt

≤ 2(4π)−d/2|x− y|

∫ ∞

0

t−d/2−1/2e−r2

32tu(t) dt .

Since u is non-increasing and varies regularly at 0 with index α2− 1, by Lemma A.1

we see that there is a constant c1 > 0 so that∫ ∞

0

t−d/2−1/2e−r2

32tu(t) dt ≤ c1r−d+1u(r2) for every r ∈ (0, 1) .

Theorem 4.2 yields

|G(x)−G(y)| ≤ c2 g(r)|x−y|

r.

HARMONIC FUNCTIONS OF LEVY PROCESSES 21

When |x− y| ≥ r2,

|G(x)−G(y)| ≤ G(x) +G(y) ≤ 2g(r)

since |x|, |y| ≥ r .

Proposition 5.2. There is a constant c > 0 such that for all R ∈ (0, 1), r ∈ (0, R2],

y ∈ BR(0) and x1, x2 ∈ BR2(0) \Br(y)

|GBR(0)(x1, y)−GBR(0)(x2, y)| ≤ cg(r)(

1 ∧ |x1−x2|r

)

.

Proof. By symmetry of the Green function,

GBR(0)(xi, y) = GBR(0)(y, xi) = G(xi − y)− Ey[G(XτBR(0)− xi)]

= G(xi − y)− Ey[G(XτBR(0)− xi)] ,

for i ∈ 1, 2 . Now the result follows from Proposition 5.1.

Proposition 5.3. There is a constant c > 0 such that for any r ∈ (0, 1) andx, y ∈ B r

8(0):

(i) if z ∈ B2r(0) \Br(0), then

∣∣KBr(0)(x, z)−KBr(0)(y, z)

∣∣ ≤ c|z|−dφ ((|z| − r)−2)

φ (|x− y|−2);

(ii) if z 6∈ B2r(0), then

∣∣KBr(0)(x, z)−KBr(0)(y, z)

∣∣ ≤ c

j(

|z|2

)

φ (|x− y|−2).

Proof. In the estimate

∣∣KBr(0)(x, z)−KBr(0)(y, z)

∣∣ ≤

∫

Br(0)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv

we split the integral into three parts:

I1 =

∫

B2|x−y|(x)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv

I2 =

∫

B r4(x)\B2|x−y|(x)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv

I3 =

∫

Br(0)\B r4(x)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv .

22 ANTE MIMICA

For the first part we obtain

I1 ≤

∫

B2|x−y|(x)

GBr(0)(x, v)j(|z − v|) dv +

∫

B3|x−y|(y)

GBr(0)(y, v)j(|z − v|) dv

≤ 2j(

|z|2

)∫

B3|x−y|(0)

G(v) dv ≤ c1j( |z|

2 )φ(|x−y|−2)

, (5.2)

for any z 6∈ Br(0). We have used Theorem 4.2 to get the last inequality in (5.2).

In order to estimate I2 we split the integral in the following way. We let N =⌊log r

4|x−y|

log 2

⌋

and write

I2 ≤

N∑

n=1

∫

B2n+1|x−y|(x)\B2n |x−y|(x)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv .

Now, for each n ∈ 1, . . . , N we can apply Proposition 5.2 (with the correspondingradii (2n − 1)|x− y| and r) to get

∫

B2n+1|x−y|(x)\B2n |x−y|(x)

∣∣GBr(0)(x, v)−GBr(0)(y, v)

∣∣ j(|z − v|) dv

≤ c3g ((2n − 1)|x− y|)

2n − 1

∫

B2n+1|x−y|(x)

j(|z − v|) dv .

By Theorem 4.2

g ((2n − 1)|x− y|)

g(|x− y|)≤ c4

η((2n − 1)|x− y|)

η(|x− y|)for all n ∈ 1, 2, . . . , N,

with η(r) = r−d−2 φ′(r−2)φ(r−2)2

.

Noting that η varies regularly at zero with index α−d < 0, the uniform convergencetheorem for regularly varying functions (see Theorem 1.5.2 in [BGT87]) gives

η ((2n − 1)|x− y|)

η(|x− y|)≤ c5(2

n − 1)α−d for all n ∈ N and |x− y| ≤ 12.

HARMONIC FUNCTIONS OF LEVY PROCESSES 23

By Theorem 4.2 and (2.9) g(|x− y|) ≤ c5φ(|x−y|−2)

and so

I2 ≤ c6

N∑

n=1

(2n − 1)α−d−1g(|x− y|)(2n+1|x− y|)dj(

|z|2

)

≤ c7j(

|z|2

)

φ (|x− y|−2)

N∑

n=1

2(α−1)n

≤c7

1− 2α−1

j(

|z|2

)

φ (|x− y|−2)for every z 6∈ Br(0) .

It remains to estimate I3. Applying Theorem 5.2 we get

I3 ≤ c8g(r)|x− y|

r

∫

Br(z)

j(|v|) dv

≤ c9|x− y|φ (|x− y|−2)

rφ (r−2)

r−d

φ (|x− y|−2)

∫

Br(z)

j(|v|) dv

≤ c10r−d

φ (|x− y|−2)

∫

Br(z)

j(|v|) dv . (5.3)

In the last inequality we have used the theorem of Potter (cf. [BGT87, Theorem1.5.6 (iii)]) to conclude that for δ < 1− α there is a constant Aδ > 0 such that

|x− y|φ(|x− y|−2)

rφ(r−2)≤ Aδ

(|x− y|

r

)1−α−δ

≤ Aδ,

since r 7→ rφ(r−2) varies regularly at zero with index 1− α .

Since

j(|v|) ≥ j(

|z|2

)

for all v ∈ Br(z) and z ∈ B2r(0)c

it follows from (5.3) that

I3 ≤ c11j(

|z|2

)

φ (|x− y|−2),

On the other hand, for z ∈ B2r(0) \Br(0) we deduce from

Br(z) ⊂ B3(0) \B|z|−r(0)

(similarly as in (4.2)) that∫

Br(z)

j(|v|) dv ≤ c12φ((|z| − r)−2

).

24 ANTE MIMICA

By (5.3)

I3 ≤ c13|z|−dφ ((|z| − r)−2)

φ (|x− y|−2)for all z ∈ B2r(0) \Br(0) .

6. Regularity of harmonic functions

Recall that (2.7) gives the representation for any bounded function f : Rd → R thatis harmonic in B2r(x0):

f(x) = Ex

[

f(

XτBr(x0)

)]

=

∫

Br(x0)cKBr(x0)(x, z)f(z) dz, x ∈ Br(x0) . (6.1)

Proof of Theorem 1.1. By (6.1)

|f(x)− f(y)| ≤ ‖f‖∞

∫

B2r(0)c

∣∣KB2r(0)(x, z)−KB2r(0)(y, z)

∣∣ dz . (6.2)

It remains to estimate the integral in (6.2), which we split in the following way

I1 =

∫

B4r(0)\B2r(0)

∣∣KB2r(0)(x, z)−KB2r(0)(y, z)

∣∣ dz

I2 =

∫

B1(0)\B4r(0)

∣∣KB2r(0)(x, z)−KB2r(0)(y, z)

∣∣ dz

I3 =

∫

B1(0)c

∣∣KB2r(0)(x, z)−KB2r(0)(y, z)

∣∣ dz

In order to estimate I1 we use Proposition 5.3 (i). More precisely,

I1 ≤c1

φ (|x− y|−2)

∫

B4r(0)\B2r(0)

|z|−dφ((|z| − 2r)−2

)dz

=c2

φ (|x− y|−2)

∫ 4r

2r

t−1φ((t− 2r)−2

)dt

≤c2

φ (|x− y|−2)(2r)−1

∫ 2r

0

φ(t−2

)dt

≤c3

φ (|x− y|−2)φ(r−2) ,

where in the last inequality we have used Karamata’s theorem (see the 0-version ofTheorem 1.5.11 in [BGT87]).

HARMONIC FUNCTIONS OF LEVY PROCESSES 25

We estimate I2 and I3 with the help of Proposition 5.3 (ii). Since the Levy measureis finite away from the origin,

I3 ≤c4

φ (|x− y|−2)

∫

B1(0)cj(

|z|2

)

dz ≤c5

φ (|x− y|−2).

Also,

I2 ≤c6

φ (|x− y|−2)

∫

B1(0)\B4r(0)

j(

|z|2

)

dz ≤c7φ (r

−2)

φ (|x− y|−2),

where in the last inequality we have used Theorem 4.1.

7. Examples

In this section is to illustrate our results by some examples.

7.1. (Iterated) Geometric stable processes. This class of examples belongs tothe case of α = 0.

Let β ∈ (0, 2]. We define a family of functions φn : (0,∞) → (0,∞) : n ∈ Nrecursively by

φ1(λ) = log(1 + λβ/2), λ > 0

φn+1 = φ1 φn, n ∈ N .

The function φ1 is a complete Bernsetin function. Since complete Bernstein functionsare closed under operation of composition, φn belongs to this class for every n ∈ N.

Let Sn be a subordinator with the Laplace exponent φn. S1 is known as the geomet-

ric β2-stable subordinator. We call Sn the iterated geometric β

2-stable subordinator.

The corresponding subordinate Brownian motions Xn will be called (iterated) geo-metric β-stable processes.

As already remarked in [SSV06], these processes show quite different behavior com-pared to the one of stable processes. Our contribution to this class of examples isthat now we can obtain behavior of the Levy density as a special case of Theorem4.1(even for iterated geometric stable processes).

26 ANTE MIMICA

The Levy density of Xn is comparable to

1

|x|d·

n−1∏

k=1

1

logk(|x|−1)

as |x| → 0+ ,

which is almost integrable. We can say that (intially) this process jumps slower thanany stable processes.

This can be also seen from the behavior of the Green function:

G(x, y) ≍1

|x− y|d log2n(|x− y|−1)·n−1∏

k=1

1

logk(|x− y|−1)as |x− y| → 0 + .

As a consequence, E0τBr(0) ≍1

logn(r−1)

as r → 0+. Therefore Xn needs (on average)

more time to exit ball Br(0) than any stable process or Brownian motion.

Theorem 1.1 implies the following a-priori local regularity estimates of harmonicfunctions:

|f(x)− f(y)| ≤ c‖f‖∞ logn(r−1)

1

logn(|x− y|−1)for all x, y ∈ B r

2(0)

and any bounded function f which is harmonic in Br(0) .

This tells us that the modulus of continuity is bounded by a logarithmic term. It isstill an open problem whether these harmonic functions satify a-priori local Holdercontinuity estimates.

7.2. Conjugates of (iterated) geometric stable processes. This class of ex-amples corresponds to the case α = 2.

Let ψn(λ) =λ

φn(λ), where φn are as in Subsection 7.1.

Since φn are complete Bernstein functions, ψn are also complete Bernstein func-tions. Therefore, there exist (killed) subordinators T n with the Laplace exponentψn. Killing will not affect the behavior of the Levy and potential densities of T n

near zero.

In this case the Levy density of the corresponding subordinate Brownian motion Y n

behaves near the origin as

1

|x|d+2 log2n(|x|−1)

·n−1∏

k=1

1

logk(|x|−1)

as |x| → 0 + .

HARMONIC FUNCTIONS OF LEVY PROCESSES 27

Note that the integrability conditions of the Levy measure are barely satisfied inthis case.

Comparing this behavior to the behavior of the small jumps of the α-stable process,we see that small jumps of Y n are more intensive.

Another interesting feature of this process is the following behavior of the Greenfunction:

G(x, y) ≍ |x− y|2−d logn(|x− y|−1) as |x− y| → 0 + .

In this sense the process Y n is ’between’ stable processes and Brownian motion,since their Green functions are given by

G(α) = cα|x− y|α−d and G(2) = cα|x− y|2−d .

Appendix A. Asymptotical properties

In the appendix we prove a technical lemma which is used throughout the paper.

Lemma A.1. Let w : (0,∞) → (0,∞) be a decreasing function satisfying

w(t) ≍ t−bℓ(t), t→ 0+ ,

for a function ℓ : (0,∞) → (0,∞) that varies slowly at 0 and b ≥ 0 .

If p > 1 and a > 0, then

I(r) =

∫ ∞

0

t−pe−art w(t) dt, r > 0 ,

satisfies

I(r) ≍ a−p−b+1r−p+1w(r), r → 0 + .

Proof. Change variables yields

I(r) = (ar)−p+1

∫ ∞

0

e−ttp−2w(ar

t

)

dt (A.1)

By assumptions, there are constants c1, c2 > 0 and r0 > 0 such that

c1a−b ≤

w(ar)

w(r)≤ c2a

−b for every r ∈ (0, r0) . (A.2)

28 ANTE MIMICA

Let us first prove the upper bound. Using the fact that w is decreasing, by (A.1)and (A.2) we get

I(r) ≤ (ar)−p+1

∫ 1

0

e−ttp−2w (ar) dt+ (ar)−p+1

∫ ∞

1

e−ttp−2w(ar

t

)

dt

≤ c2(ar)−p+1a−bw(r)

[∫ 1

0

e−ttp−2 dt+

∫ ∞

1

e−ttp+b−2 dt

]

≤ c′2a−p−b+1r−p+1w(r)

for every r ∈ (0, r0) .

The lower bound follows similarly:

I(r) ≥ (ar)−p+1

∫ ∞

1

e−ttp−2w (ar) dt ≥ c1(ar)−p+1a−bw(r)

∫ ∞

1

e−ttp−2 dt

= c′1a−p−b+1r−p+1w(r)

for every r ∈ (0, r0) .

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Fakultat fur Mathematik, Universitat Bielefeld, Germany

E-mail address : amimica@math.uni-bielefeld.de